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[Math.QA] 24 Jan 2003 Ihagbacsprrushwvr Hr R Nsm Sense Some in Are There However, Supergroups Algebraic with En Anyte Pwt Hsclproblems

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[Math.QA] 24 Jan 2003 Ihagbacsprrushwvr Hr R Nsm Sense Some in Are There However, Supergroups Algebraic with En Anyte Pwt Hsclproblems ON ALGEBRAIC SUPERGROUPS AND QUANTUM DEFORMATIONS R. FIORESI* Dipartimento di Matematica, Universita’ di Bologna Piazza Porta San Donato 5, 40126 Bologna, Italy e-mail: fi[email protected] Abstract. We give the definitions of affine algebraic supervariety and affine algebraic supergroup through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroups GL(m|n) and SL(m|n) and give explicitly the Hopf algebra structure of the algebra representing the functors of points. At the end we give also the quantization of GL(m|n) together with its coaction on suitable quantum spaces according to Manin’s philosophy. 1. Introduction The mathematical foundations of supergeometry were laid in the 60s by Berezin in [Be] and later by Leites [Le] Kostant [Ko] and Manin [Ma1] among many others, its origins being mainly tied up with physical problems. A new attention to the subject came later with the study of quantum fields and superstring. In the 1999 “Notes on Supersymmetry” Deligne and Morgan [DM] give a categorical point of view on supersymmetry notions developed originally by physicists and known from a more “operational” point of view. In the current definitions of supermanifold, the points of a supermanifold are points of an usual manifold and the adjective super refers to an additional structure on the structural sheaf of functions on the manifold. This sheaf is assumed to be a sheaf of commutative superalgebras, where a superalgebra is a Z2-graded algebra. When dealing with algebraic supergroups however, there are in some sense true points. In fact, for each arXiv:math/0111113v2 [math.QA] 24 Jan 2003 supercommutative algebra A the A points of a supergroup can be viewed as a certain subset of automorphisms of a superspace Am|n where A is a given commutative superalgebra. For this reason the point of view of functor of points is in this case most useful. It allows to associate to an affine supergroup a commutative Hopf superalgebra the same way it does in the commutative case. It is hence possible to define a quantum deformation of an algebraic supergroup in complete analogy to the non super case: it will be a deformation of the commutative Hopf superalgebra associated to it. * Investigation supported by the University of Bologna, funds for selected research topics. 1 The quantum supergroup GL(m|n) was first constructed by Manin [Ma3] together with its coactions on suitable quantum superspaces. The construction of its Hopf superal- gebra structure is in some sense implicit. In a subsequent work, Lyubashenko and Sudbery [LS] provided quantum deformations of supergroups of GL(m|n) type using the universal R-matrix formalism. The same formalism is also used in [P] where more explicit formu- las are given. In both works however the ring and its coalgebra structure do not appear explicitly, since the calculation was too involved using the R-matrix approach. In the present paper we present a definition of affine algebraic supervariety and su- pergroup that is basically equivalent to the one of Manin [Ma1] and then we use this point of view to give a quantum deformation of the supergroup GL(m|n). The main result of the paper consists in giving explicitly the coalgebra structures for the Hopf superalgebras associated to the supergroup GL(m|n) and its quantization kq[GL(m|n)] obtained according to the Manin phylosophy, that is together with coactions on suitable quantum spaces. The explicit forms of the comultiplication for GL(m|n) and its quantization kq[GL(m|n)] are non trivial, they rely heavily on the presence of the nilpotents of the superalgebras and do not appear in any other work. These results were announced in the proceeding [Fi] where they appeared without proof. The organization of this paper is as follows. In §2 we introduce the notion of affine supervariety and affine supergroup using the functor of points. These two definitions turn out to be basically equivalent to the definitions that one finds in the literature ([De], [Ma1] among many others). In §3 we write explicitly the Hopf algebra structure of the Hopf superalgebra associated to the supergroups GL(m|n) and SL(m|n), where with GL(m|n) we intend the supergroup whose A points are the group of automorphisms of the superspace Am|n and with SL(m|n) the subsupergroup of GL(m|n) of automorphisms with Berezinian equal to 1, with A commutative superalgebra. In §4 we construct the non commutative Hopf superalgebra kq[GL(m|n)] deformation in the quantum group sense of the Hopf superalgebra associated to GL(m|n). We also see that kq[GL(m|n)] admits coactions on suitable quantum superspaces. The author wishes to thank Prof. V. S. Varadarajan, Prof. I. Dimitrov, Prof. A. Vistoli, Prof. A. Sudbery, Prof. T. Lenagan and Prof. M. A. Lledo for helpful comments. 2. Preliminaries on algebraic supervarieties and supergroups Let k be an algebraically closed field. All algebras and superalgebras have to be intended over k unless otherwise specified. Given a superalgebra A we will denote with A0 A the even part, with A1 the odd part and with Iodd the ideal generated by the odd part. 2 A superalgebra is said to be commutative (or supercommutative) if xy =(−1)p(x)p(y)yx, for all homogeneous x,y where p denotes the parity of an homogeneous element (p(x)=0if x ∈ A0, p(x)=1 if x ∈ A1). In this section all superalgebras are assumed to be commutative. Let’s denote with A the category of affine superalgebras that is commutative super- algebras such that, modulo the ideal generated by their odd part, they are affine algebras (an affine algebra is a finitely generated reduced commutative algebra). Definition (2.1). Define affine algebraic supervariety over k a representable functor V from the category A of affine superalgebras to the category S of sets. Let’s call k[V ] the commutative k-superalgebra representing the functor V , V (A) = Homk−superalg(k[V ],A), A ∈A. We will call V (A) the A-points of the variety V . A morphism of affine supervarieties is identified with a morphism between the repre- senting objects, that is a morphism of affine superalgebras. We also define the functor Vred associated to V from the category Ac of affine k- algebras to the category of sets: k[V ] Vred(Ac) = Homk−alg(k[V ]/Iodd ,Ac), Ac ∈Ac. Vred is an affine algebraic variety and it is called the reduced variety associated to V . If the algebra k[V ] representing the functor V has the additional structure of a com- mutative Hopf superalgebra, we say that V is an affine algebraic supergroup. (For the definition and main properties of Hopf algebras see [Mo]). Remarks (2.2). 1. Let G be an affine algebraic supergroup in the sense of (2.1). As in the classical setting, the condition k[G] being a commutative Hopf superalgebra makes the functor group valued, that is the product of two morphisms is still a morphism. In fact let A be a commutative superalgebra and let x,y ∈ Homk−superalg(k[G],A) be two points of G(A). The product of x and y is defined as: x · y =def mA · x ⊗ y · ∆ where mA is the multiplication in A and ∆ the comultiplication in k[G]. One can directly check that x · y ∈ Homk−superalg(k[G],A), that is: (x · y)(ab)=(x · y)(a)(x · y)(b) 3 This is an important difference with the quantum case that will be treated in §4. The non commutativity of the Hopf algebra in the quantum setting does not allow to multiply morphisms(=points). In fact in the quantum (super)group setting the product of two morphisms is not in general a morphism. For more details see [Ma4] pg 13. 2. Let V be an affine algebraic supervariety as defined in (2.1). Let k0 ⊂ k be a subfield of k. We say that V is a k0-variety if there exists a k0-superalgebra k0[V ] such ∼ that k[V ] = k0[V ] ⊗k0 k and V (A) = Homk0−superalg(k0[V ],A) = Homk−superalg(k[V ],A), A ∈A. We obtain a functor that we still denote by V from the category Ak0 of affine k0-superalgebras to the category of sets: V (Ak0 ) = Homk0−superalg(k0[V ],Ak0 ), A ∈Ak0 . This allows to consider rationality questions on a supervariety. We will not pursue this further in the present work. Examples (2.3). 1. The k-points of an affine supervariety V correspond to the affine variety defined over k whose functor of points is Vred. 2. Let A be a commutative superalgebra. Let M(m|n)(A) be the linear endomor- phisms of the superspace Am|n (see [De] pg. 53): a11 ... a1,m α1,m+1 ... α1,m+n . . am, ... am,m αm,m ... αm,m n 1 +1 + αm , ... αm ,m am ,m ... am ,m n +1 1 +1 +1 +1 +1 + . . α ... α a ... a m+n,1 m+n,m m+n,m+1 m+n,m+n aij ∈ A0, αkl ∈ A1, 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n, 1 ≤ k ≤ m, m +1 ≤ l ≤ m + n or m +1 ≤ k ≤ m + n, 1 ≤ l ≤ m. This is an affine supervariety represented by the commutative superalgebra: k[M(m|n)] = k[xij,ξkl] where xij’s and ξkl’s are respectively even and odd variables with 1 ≤ i, j ≤ m or m +1 ≤ i, j ≤ m + n,1 ≤ k ≤ m, m +1 ≤ l ≤ m + n or m +1 ≤ k ≤ m + n, 1 ≤ l ≤ m.
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